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Mirrors > Home > MPE Home > Th. List > Mathboxes > mexval | Structured version Visualization version GIF version |
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mexval.k | β’ πΎ = (mTCβπ) |
mexval.e | β’ πΈ = (mExβπ) |
mexval.r | β’ π = (mRExβπ) |
Ref | Expression |
---|---|
mexval | β’ πΈ = (πΎ Γ π ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mexval.e | . 2 β’ πΈ = (mExβπ) | |
2 | fveq2 6892 | . . . . . 6 β’ (π‘ = π β (mTCβπ‘) = (mTCβπ)) | |
3 | mexval.k | . . . . . 6 β’ πΎ = (mTCβπ) | |
4 | 2, 3 | eqtr4di 2791 | . . . . 5 β’ (π‘ = π β (mTCβπ‘) = πΎ) |
5 | fveq2 6892 | . . . . . 6 β’ (π‘ = π β (mRExβπ‘) = (mRExβπ)) | |
6 | mexval.r | . . . . . 6 β’ π = (mRExβπ) | |
7 | 5, 6 | eqtr4di 2791 | . . . . 5 β’ (π‘ = π β (mRExβπ‘) = π ) |
8 | 4, 7 | xpeq12d 5708 | . . . 4 β’ (π‘ = π β ((mTCβπ‘) Γ (mRExβπ‘)) = (πΎ Γ π )) |
9 | df-mex 34478 | . . . 4 β’ mEx = (π‘ β V β¦ ((mTCβπ‘) Γ (mRExβπ‘))) | |
10 | fvex 6905 | . . . . 5 β’ (mTCβπ‘) β V | |
11 | fvex 6905 | . . . . 5 β’ (mRExβπ‘) β V | |
12 | 10, 11 | xpex 7740 | . . . 4 β’ ((mTCβπ‘) Γ (mRExβπ‘)) β V |
13 | 8, 9, 12 | fvmpt3i 7004 | . . 3 β’ (π β V β (mExβπ) = (πΎ Γ π )) |
14 | xp0 6158 | . . . . 5 β’ (πΎ Γ β ) = β | |
15 | 14 | eqcomi 2742 | . . . 4 β’ β = (πΎ Γ β ) |
16 | fvprc 6884 | . . . 4 β’ (Β¬ π β V β (mExβπ) = β ) | |
17 | fvprc 6884 | . . . . . 6 β’ (Β¬ π β V β (mRExβπ) = β ) | |
18 | 6, 17 | eqtrid 2785 | . . . . 5 β’ (Β¬ π β V β π = β ) |
19 | 18 | xpeq2d 5707 | . . . 4 β’ (Β¬ π β V β (πΎ Γ π ) = (πΎ Γ β )) |
20 | 15, 16, 19 | 3eqtr4a 2799 | . . 3 β’ (Β¬ π β V β (mExβπ) = (πΎ Γ π )) |
21 | 13, 20 | pm2.61i 182 | . 2 β’ (mExβπ) = (πΎ Γ π ) |
22 | 1, 21 | eqtri 2761 | 1 β’ πΈ = (πΎ Γ π ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 Vcvv 3475 β c0 4323 Γ cxp 5675 βcfv 6544 mTCcmtc 34455 mRExcmrex 34457 mExcmex 34458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-mex 34478 |
This theorem is referenced by: mexval2 34494 msubff 34521 msubco 34522 msubff1 34547 mvhf 34549 msubvrs 34551 |
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